By Shiming Duan, Last Update: 8/11/2010 3:03 PM
III. Stability of Systems of DDEs via the Lambert W Function
One advantage for the Lambert W function based approach is that it enables us to locate the poles of the principal branch and use the result to determine the stability of a system of DDEs. For example, we can use this method to design the control of DDEs with guarrenteed closed-loop stability and performance. Here we summarize the results of the stability analysis of DDEs using the Lambert W function.
Scalar Case
It has been proven in (H. Shinozaki and T. Mori, 2006) that, for the scalar DDEs, the rightmost pole in the s-plane is determined by the principal branch () of . Thus,
One can use only the principal branch to determine the stability of scalar DDEs.
Matrix Case
The proof for the scalar case can readily be extended to the matrix case when and are simultaneously triangularizable (Radjavi and Rosenthal, 2000). No such proof is currently available for the general case of matrix DDEs. However, in all the examples considered in the literature, it has been observed that the rightmost eigenvalue is obtained using the first m branches, where m is the nullity of . We state this here as a generalization of the conjecture in the book (see section 3.3.1 in the book):
where .
When has no repeated zero eigenvalues:
Thus, if has no repeated zero eigenvalues, or has full rank, one can use only the principal branch to determine the stability of matrix DDEs. Otherwise, all the first m branches must be examined to yield a conclusion.
Example 3.1: Consider the pendulum system in example 1.2. Let l=2m, m=1kg, g=10ms-2 , and =1s. The control is a velocity feedback control . The physical meaning of the control is to add delayed damping to stabilize the system and the objective is to determine
the proper coefficient c for the controller so that the closed-loop system described in the following equation is stable. Further, we are interested in the free response of the closed-loop system
under a certain initial condition .
Step 1. First, check the rank of. In this example, has a rank of one if since . Thus, we can use only the principal branch to check the stability of the DDE. If , the DDE becomes an ODE and one can verify that the system is not asymptotically stable since . Since one of the eigenvalues of is zero, the regular approach (see Ex. 2.2) cannot be used because . Thus, one must apply the hybrid branch approach to solve the problem.
Step 2. Solve Q0,0 for the principal branch using the fsolve function.
Let and we have
One can verify that the Jordan canonical form of is
The matrix Lambert W function with hybrid branches are defined as
.
Thus, Q0,0 satisfies
Step 3. After is obtained, we can compute from the equation . Based on the conjecture, we can determine the stability of the system based on . One can verify, using either the Lambert W function approach or Matlab dde23 function, that the closed-loop system is stable when . Here we pick and obtain . The table below lists the results for different values of c.
C |
c = -2 |
c = 2 |
c = 4 |
c = 6 |
c = 8 |
Q0,0 |
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S0,0 |
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Step 4. After c is selected, we need to obtain the solution of the closed-loop DDE. For branches other than the principal branch, we must use the matrix Lambert W function with hybrid branches, which is defined as
.
By setting k2=0 and letting k1 vary from to , we can solve Eq. (2.15) in book to obtain . After is obtained, we can compute from the equation . The following table shows the results for part of the branches.
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Eigenvalues of |
k1=0 & k2=0 |
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k1=-1 & k2=0 |
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k1=1 & k2=0 |
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k1=-2 & k2=0 |
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k1=2 & k2=0 |
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Remark:
When Ad does not have full rank and the hybrid branch approach is applied, it is possible to find more than one solution for each , the non-principal branches. One may find different for a certain branch by using different initial conditions to initialize the fsolve function to solve the equation
In this example, k2 = 0 and k1 =,...,-1, 0, 1, ...,. From the results, we observe that the eigenvalues in the principal branch are repeated in the non-principal branches, which arises due to by fixing k2 = 0 when calculating the hybrid matrix Lambert W function. We also note that some eigenvalues from the non-principal branches appear in other non-principal branches, which also makes this case very special. Because of these special features, for the rest of the steps, it is more convenient to find the solution using the following procedure instead of the general one introduced in Ex. 2.2.
For the hybrid branch case (rank(Ad)<n), one should
* Find the set of distinct eigenvalues , j =,...,-1, 0, 1, ...,, from the results for . Note here are scalars instead of matrices. In this example, the set of distinct eigenvalues is .
* Use approaches similar to partial fraction expansion to find coefficients for each (see Step 5 below)
* The solution of the free response can then be obtained as
, where and
Note here and are vectors with same dimension and are scalar functions of time. The multiplicity of can be determined by evaluating the limit
The smallest natural number q satisfying the above equation is the multiplicity of
Step 5. To compute , one should solve the partial fraction expansion for this infinite series. When the multiplicity of , the solution can be formulated as
and the can be determined as
For the case when the multiplicity of , one can take the derivative of the numerator and denominator first before evaluating the limit (similar to the PFE for a ratio of two polynomials). For example, when has multiplicity of 2 while the other distinct eigenvalues have multiplicity of 1:
Then
The table below lists the results for under the initial condition
Index j |
Eigenvalue |
Multiplicity of |
Coefficient |
1 |
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1 |
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2 |
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1 |
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3 |
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1 |
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4 |
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1 |
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5 |
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1 |
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6 |
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1 |
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7 |
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1 |
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8 |
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1 |
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Thus, the free response of the matrix DDE system in this example can be determined as
and can be approximated by summing a finite number of branches. The solution obtained using the Lambert W function approach for the first 8 distinct eigenvalues is compared with the simulation results using the Matlab dde23 function in Fig.3.1.
Fig. 3.1 Comparison between the solution obtained using the Lambert W function method for the first 8 distinct eigenvalues (solid) to the results from the Matlab dde23 function (dash) for Example 3.1